Problem: $ 3.\overline{5} \div 0.\overline{17} = {?} $
Answer: First convert the repeating decimals to fractions. $\begin{align*} 10x &= 35.5555...\\ x &= 3.5555...\end{align*} $ $\begin{align*} 9x &= 32 \\ x &= \dfrac{32}{9}\end{align*} $ $\begin{align*} 100y &= 17.1717...\\ y &= 0.1717...\end{align*} $ $\begin{align*} 99y &= 17 \\ y &= \dfrac{17}{99}\end{align*} $ So, the problem becomes: $ \dfrac{32}{9} \div \dfrac{17}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{32}{9} \times \dfrac{99}{17} = {?} $ $ \phantom{\dfrac{32}{9} \times \dfrac{17}{99}} = \dfrac{32 \times 99}{9 \times 17} $ $ \phantom{\dfrac{32}{9} \times \dfrac{17}{99}} = \dfrac{32 \times \cancel{99}11} {\cancel{9} \times 17} $ $ \phantom{\dfrac{32}{9} \times \dfrac{17}{99}} = \dfrac{352}{17} $